I was called to the body shop 2 weeks ago. I finally was able to go today. I knew the right rear quarter panel had some work done to it from some photo's the PO gave me.
There was some fiberglass added to the inside of the quarter panels to hide RUST. Well I bought the car with tunnel vision not knowing exactly what to look for. So if you are reading this thread and are looking at a old Mustang don't trust anyone get it professionally inspected. So you don't get burned like me. I have the photos that prove the car was disguised to hide the rust. I only got them well after the sale. What I'm saying is get the car car any car inspected prior to buying it. Or get a second opinion.

I'm trying to remove the heater core and the door is in the way. I removed the door rod cap from one side, but how does it come out on the other side? I'm restoring the evaporator core myself. So does the door need to be removed to get it out?

I'm assuming this is a factory A/C unit?
You should remove the COMPLETE heater/AC unit as unit. When it is out of the car then open the fiberglass casing. The 2 halves (upper & lower) are held together with spring clips. This will expose the heater core & evaporator, plus the various 'doors' & seals inside the box.
First step is to remove the 2 'pie plates' where the heater lines & freon lines protrude through the firewall.
Second step is to remove the passenger seat (you'll need the room).
Third- disconnect the vacuum lines & electrical connectors.
The entire unit is secured in the car by studs through the firewall and another fastener up by the cowl fresh air inlet. Be very careful in the removal. The studs on the fiberglass housing can be broken. IIRC, the mating of the unit to the blower motor is just a slip joint.

This is NOT a fun job. Be patient and go slow.
Be SURE to pressure check the new heater core prior to installing. (Ask me how I know)
I've never found a bad evaporator in '67 or '68. Flushing with an A/C cleaning solvent should do the trick. But as long as you have access- leak check it as well.
Installing the completed unit is a PITA- lining up all the studs through the firewall and securing the fastener at the Fresh Air inlet requires a helper.

I converted my CS to a rotary compressor & R134 freon, with a new expansion valve & oversized Condensor + hoses from ACC. It blows out 40 degree air.
Hope all this helps. If you need me: 828-507=2530 cell

Its been out. I should've said it was already out. Sorry bout that. I posted a photo update. The part with my hand on it. Does that need to be removed to get the core out? So the evaporator does not need to be replaced if it is leak free? I'm going to get it checked out, but would you replace the heater core since its already out just to be on the safe side?

Since you have it all opened up, it would be wise to replace the heater core. Be sure you get a COPPER unit, and still leak check it.
Not a lot of wiggle room to remove the core. The part that your hand is on may have to come out to give space to remove the core.

I finally was able to bring the evaporator core to the radiator shop. They freed the door and removed the heater core and cleaned it and checked it out along with the evaporator coil. They were all ok and it was only $85. So I think it was a good deal. I'm going to get it cleaned up and change the gaskets. Anybody know what is a good paint to paint the case? I started working on the 70 gas tank and have to drill some holes in the tank armor. Trying to get everything together to put the engine back in.

I had to move the motor Monday. The timing cover was glass bead blasted by the engine shop and left unpainted. Should I paint it or leave it unpainted? I can't see it unpainted and the water pump painted.